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I was taught that a probit with fixed effects would not be consistent because the estimates of a non-linear model with a link function other than the canonical (in this case the logit) are not sufficient statistics. Among other things, Stata will not allow for a fixed effects probit (or clog-log, as a matter of fact).

Why would the consistency of the fixed effects estimator depend on the link function not being the canonical one?

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  • $\begingroup$ Related to this question, perhaps you will find the answers in there useful: stats.stackexchange.com/questions/55316/… $\endgroup$ – Andy Mar 25 '15 at 21:42
  • $\begingroup$ It is a different question, but the notes you linked to in your answer seem promising. I still do not understand why the conditional fixed effects cannot be estimated with a link function other than the logit, though. $\endgroup$ – Kenji Mar 25 '15 at 22:22

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